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Answer :
We start by converting Shaina’s height into feet. Since she is 5 feet 6 inches tall and there are 12 inches in a foot, we have:
[tex]$$
\text{Shaina's height} = 5 + \frac{6}{12} = 5 + 0.5 = 5.5 \text{ feet.}
$$[/tex]
Next, note that she stands 20 feet from the tree. When she looks upward at an angle of elevation of [tex]$68^\circ$[/tex], she is essentially forming a right triangle where:
- The horizontal side (adjacent side) is 20 feet.
- The vertical side (opposite side) is the extra height from her eyes to the top of the tree.
- The angle at her position is [tex]$68^\circ$[/tex].
Let the extra height above her height be denoted by [tex]$h_{\text{extra}}$[/tex]. From trigonometry, we know:
[tex]$$
\tan(68^\circ) = \frac{h_{\text{extra}}}{20}.
$$[/tex]
Solving for [tex]$h_{\text{extra}}$[/tex], we have:
[tex]$$
h_{\text{extra}} = 20 \tan(68^\circ).
$$[/tex]
Evaluating [tex]$20 \tan(68^\circ)$[/tex] (using a calculator or trigonometric tables), we find:
[tex]$$
20 \tan(68^\circ) \approx 49.50 \text{ feet.}
$$[/tex]
The total height of the tree is the sum of Shaina’s height and the extra height:
[tex]$$
\text{Tree's height} = 5.5 + 49.50 = 55 \text{ feet.}
$$[/tex]
Thus, the height of the tree is [tex]$\boxed{55~\text{feet}}$[/tex].
[tex]$$
\text{Shaina's height} = 5 + \frac{6}{12} = 5 + 0.5 = 5.5 \text{ feet.}
$$[/tex]
Next, note that she stands 20 feet from the tree. When she looks upward at an angle of elevation of [tex]$68^\circ$[/tex], she is essentially forming a right triangle where:
- The horizontal side (adjacent side) is 20 feet.
- The vertical side (opposite side) is the extra height from her eyes to the top of the tree.
- The angle at her position is [tex]$68^\circ$[/tex].
Let the extra height above her height be denoted by [tex]$h_{\text{extra}}$[/tex]. From trigonometry, we know:
[tex]$$
\tan(68^\circ) = \frac{h_{\text{extra}}}{20}.
$$[/tex]
Solving for [tex]$h_{\text{extra}}$[/tex], we have:
[tex]$$
h_{\text{extra}} = 20 \tan(68^\circ).
$$[/tex]
Evaluating [tex]$20 \tan(68^\circ)$[/tex] (using a calculator or trigonometric tables), we find:
[tex]$$
20 \tan(68^\circ) \approx 49.50 \text{ feet.}
$$[/tex]
The total height of the tree is the sum of Shaina’s height and the extra height:
[tex]$$
\text{Tree's height} = 5.5 + 49.50 = 55 \text{ feet.}
$$[/tex]
Thus, the height of the tree is [tex]$\boxed{55~\text{feet}}$[/tex].
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